Retargeting animation with gizmos, delete gizmos in action editor or export without them
After many different approaches I found a solution myself.

To seperate the animations you want, you have to put them in NLA tracks and when exporting to .fbx you must go to animations and uncheck all actions:



It will only export the animations from the NLA tracks

To add animations to NLA tracks you have to open dope sheet -> action editor in one window and NLA Editor in the other. Then you have to select each animation which you want to keep one by one in the action editor and add it to an NLA track on the NLA editor. You have to add them to the armature not the mesh.

Here is how it has to look: